Re: equality is beautiful

[Michael Shulman <shulman@uchicago.edu>]

To rephrase what Toby said: the construction of limits via products and
equalizers only works for limits over a domain category which has a
set(oid) of objects (what Toby calls a "strict category"), whether that
set is large or small.  Of course, it also only works when the products
and equalizers exist; a particular limit may exist without the relevant
products and equalizers existing.

It is true that we are most interested in limits over small strict
categories, but there are many limits over large categories that do
exist and are occasionally useful.  For instance, every object X of a
category C is the limit of the identity functor C --> C weighted by the
representable weight hom_C(X,-), whether or not C is small or strict.
Of course, by Freyd's theorem, a large category cannot have large
*products* unless it is a preorder, so one shouldn't expect to be able
to construct such large limits via products and equalizers.  This
matches quite nicely with the type-theoretic philosophy, according to
which the large categories which arise in nature are rarely strict, so
that it wouldn't even make sense to ask for the relevant products and
equalizers.

Mike

Toby Bartels wrote:
> David Leduc wrote:
>
>> What about the characterization of limits in terms of products and
>> equalizers? It states that the limit of a functor F:J->C is
>> constructed by products indexed by the set(oid) of objects and the
>> set(oid) of arrows of J. But if you don't allow equality on objects in
>> J, you only have a preset of objects, not a set(oid).
>
> Consider the analogy between small and strict categories.
> (A category is strict if it is equipped with a notion of equality of objects.
> Logically, this is a structure rather than a property like smallness.)
>
> Often when speaking of small categories, one speaks relative to a universe
> which is a collection of set(oid)s or a collection of set(oid) cardinalities,
> so every small preset automatically comes equipped with a notion of equality.
> In this case, every small category is strict.  Conversely,
> every strict category is small relative to some universe,
> if you accept an axiom such as Grothendieck's axiom of universes.
> So these are very closely related concepts.
>
> As is well known, we are most interested in the limit of F: J -> C for small J.
> Less well known, but also true, we are most interested in it when J is strict.
> In that case, there is no problem; the arrows of J form a set(oid),
> and we can consider products indexed by that set(oid).
> In principle, J doesn't have to be strict, any more than it has to be small,
> but if you have some reason to believe that the limit exists,
> then you can examine that reason to see what product is relevant.
> Most of the time, you can just assume that J is small and strict.
>
>
> --Toby
>

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